Best Known Bounds

Along with the database of best known quantum codes in the previous section, there is also a database of best known upper and lower bounds on the maximal possible minimum weights of quantum codes. The upper bounds are not currently known with much accuracy, while the lower bounds match the minimum weights of the best known quantum codes database.

QECCLowerBound(F, n, k) : FldFin, RngIntElt, RngIntElt -> RngIntElt
Return the best known lower bound on the maximal minimum distance of [[n, k]] quantum codes over F. The bounds are currently available for binary quantum codes (which corresponds to F = GF(4)) up to length 35.
QECCUpperBound(F, n, k) : FldFin, RngIntElt, RngIntElt -> RngIntElt
Return the best known upper bound on the minimum distance of [[n, k]] quantum codes over F. The bounds are currently available for binary quantum codes (which corresponds to F = GF(4)) up to length 35.

Example QECC_QECCBounds (H167E26)

The best known lower bound on the minimum weight will always correspond to the best known quantum code from the Magma database. In this example the first code is in fact optimal, while the second one does not meet the upper bound, and so there is a theoretical possibility of an improvement. 
> F<w> := GF(4);
> Q1 := QECC(F, 20, 10);
> Q1:Minimal;
[[20, 10, 4]] Quantum code over GF(2^2)
> QECCLowerBound(F, 20, 10);
4
> QECCUpperBound(F, 20, 10);
4
>
> Q2 := QECC(F, 25, 13);
> Q2:Minimal;
[[25, 13, 4]] Quantum code over GF(2^2)
> QECCLowerBound(F, 25, 13);
4
> QECCUpperBound(F, 25, 13);
5
V2.28, 13 July 2023